Consider the graph of the parabola f(x)equals=x squaredx2. For xgreater than>0 and hgreater than>0, the secant line through (x,f(x)) and (xplus+h,f(x+h)) always has a greater slope than the tangent line at (x,f(x)).

Question

Consider the graph of the parabola f(x)equals=x squaredx2. For xgreater than>0 and hgreater than>0, the secant line through (x,f(x)) and (xplus+h,f(x+h)) always has a greater slope than the tangent line at (x,f(x)).

Choose the correct answer below.
A.
The statement is true. Since the slope of x squaredx2 is increasing for xgreater than>0, the average rate of growth on the interval [x,xplus+h] is greater than the instantaneous rate of growth at x.
B.
The statement is true. The slope of the secant line through (x,f(x)) and (xplus+h,f(xplus+h)) is left parenthesis x plus h right parenthesis squared(x+h)2, which is greater than x squaredx2, the slope of the tangent line at (x,f(x)).
C.
The statement is false. The tangent line at (x,f(x)) is the limit of secant lines through (x,f(x)) and (xplus+h,f(xplus+h)) as h goes to zero.
D.
The statement is false. If x and h both equal 1, then the tangent line at point (1,1) and the secant line through (1,1) and (2,4) have the same slope.

Answer 1

while the tangent line is the limit of the secants lines, that was not the question. It said h > 0.

Draw a parabola. The secants are always steeper than the tangents.

So, (A)

Answer 2

D. The statement is false. If x and h both equal 1, then the tangent line at point (1,1) and the secant line through (1,1) and (2,4) have the same slope.

Answer 3

The correct answer is C.

The statement is false because the tangent line at (x, f(x)) is the limit of secant lines through (x, f(x)) and (x+h, f(x+h)) as h goes to zero. As h approaches zero, the secant line becomes closer to the tangent line, and their slopes become equal at the point (x, f(x)). Therefore, there is a point where the slopes of the secant line and the tangent line are equal.

Answer 4

To determine whether the statement is true or false, let's consider the given information about the parabola f(x) = x^2.

The slope of the secant line passing through (x, f(x)) and (x + h, f(x + h)) is given by the formula:

Secant slope = (f(x + h) - f(x)) / (x + h - x)

Simplifying this expression gives:

Secant slope = (f(x + h) - f(x)) / h

Now, let's calculate the slope of the secant line using this formula.

f(x + h) = (x + h)^2
= x^2 + 2hx + h^2

So, the slope of the secant line becomes:

Secant slope = [(x^2 + 2hx + h^2) - x^2] / h
= (2hx + h^2) / h
= 2x + h

Now, let's consider the slope of the tangent line at point (x, f(x)). The slope of the tangent line is equal to the derivative of f(x) at that point, which is given by:

Tangent slope = f'(x)

Taking the derivative of f(x) = x^2:

f'(x) = 2x

Now, it's clear that the secant slope (2x + h) is greater than the tangent slope (2x) for any positive value of h. Therefore, the statement is true.

The correct answer is A. The statement is true because the slope of f(x) = x^2 is increasing for x > 0, making the average rate of growth on the interval [x, x + h] greater than the instantaneous rate of growth at x.

Consider the graph of the parabola ​f(x)equals=x squaredx2. For xgreater than>0 and hgreater than>​0, the secant line through​ (x,f(x)) and ​(xplus+​h,f(x+h)) always has a greater slope than the tangent line at​ (x,f(x)).

while the tangent line is the limit of the secants lines, that was not the question. It said h > 0.

D. The statement is false. If x and h both equal 1, then the tangent line at point (1,1) and the secant line through (1,1) and (2,4) have the same slope.

The correct answer is C.

To determine whether the statement is true or false, let's consider the given information about the parabola f(x) = x^2.

Consider the graph of the parabola f(x)equals=x squaredx2. For xgreater than>0 and hgreater than>0, the secant line through (x,f(x)) and (xplus+h,f(x+h)) always has a greater slope than the tangent line at (x,f(x)).